The derivative of f(x) at x= -3 is:
[tex]f'(-3)=\dfrac{-7}{9}[/tex]
The function f(x) is given by:
[tex]f(x)=\dfrac{-7}{x}[/tex]
We are asked to find the derivative of x at x= -3
We know that the derivative of f(x) at x=a is calculated by using the formula:
[tex]f'(a)= \lim_{h \to 0} \dfrac{f(a+h)-f(a)}{h}[/tex]
i.e.
[tex]f'(-3)= \lim_{h \to 0} \dfrac{\dfrac{7}{-3+h}-\dfrac{7}{-3}}{h}\\\\i.e.\\\\f'(-3)=\lim_{h \to 0} \dfrac{\dfrac{7\times (-3)-7\times (h-3)}{(-3)(-3+h)}}{h}\\\\f'(-3)=\lim_{h \to 0} \dfrac{-21-7h+21}{h(h-3)(-3)}\\\\f'(-3)=\lim_{h \to 0} \dfrac{-7h}{h(h-3)(-3)}\\\\f'(-3)=\lim_{h \to 0} \dfrac{-7}{(h-3)(-3)}\\\\f'(-3)=\dfrac{-7}{9}[/tex]
